Consider the Gibson. Cousin to the martini, its only distinguishing characteristic is the use of a cocktail onion as its garnish. A great cocktail onion is crisp and carries a hint of other flavors, beyond just the piquant onion. Cocktail onions, being small, are generally sweeter and less bracing than their full-sized counterparts. When you buy cocktail onions in the store, you have several choices. But many of them include additives and preservatives that would embalm a horse.
Making your own is easy. At least, it's as easy as pickling, which—let's face it—isn't terribly challenging. Make a brine, very briefly cook the onions in it, bring them down to room temperature, and place onions and brine into jars.
The worst bit of trouble you might have is finding suitable onions. Small pearl onions are ideal, but I cannot always find fresh pearl onions at my grocery. When I can find them, they're often larger than I prefer, sometimes approaching the size of a walnut or even a golf ball. Those are just too damn big. They could work in a bloody mary, I suppose, but they're way too much for a Gibson.
At those times, I fall back on the freezer section. Frozen pearl onions are always the right size for a cocktail, and they have another enormous advantage: you don't have to peel them.
In the end, it's up to you. What you use will depend on what you can find, and how much work you're willing to put in. I generally prefer to use fresh, not frozen, when I can find suitable varieties. I find that the flavor is fresher, and the onions are crisper and crunchier.
I'll be honest. I don't always have pickled onions on hand. Often, I borrow a trick from Ernest Hemingway, who was said to garnish his Gibsons with plain frozen onions. At least they help to keep your drink cold.